What is the general term of the series $1-\frac{x^{5}}{5 !}+\frac{x^{10}}{10 !}-\frac{x^{15}}{15 !}+\cdots$ ?
A) $\frac{(-1)^{k} x^{5 k}}{(5 k) !}$ for $k \geq 0$
B) $\frac{(-1)^{k+1} x^{5 k}}{(5 k) !}$ for $k \geq 0$
C) $\frac{(-1)^{k} x^{5 k+5}}{(5 k+5) !}$ for $k \geq 0$
D) $\frac{(-1)^{k+1} x^{5 k+1}}{(5 k+1) !}$ for $k \geq 0$

Question

Quizplus · Accepted Answer

The general term of the series is $\frac{(-1)^{k} x^{5 k}}{(5 k) !}$ for $k \geq 0$, which correctly represents the pattern of alternating signs, powers of $x$ increasing by multiples of 5, and factorial denominators also increasing by multiples of 5.

What Is the General Term of the Series 1−x55!+x1010!−x1515!+⋯1-\frac{x^{5}}{5 !}+\frac{x^{10}}{10 !}-\frac{x^{15}}{15 !}+\cdots1−5!x5​+10!x10​−15!x15​+⋯

What Is the General Term of the Series $1-\frac{x^{5}}{5 !}+\frac{x^{10}}{10 !}-\frac{x^{15}}{15 !}+\cdots$